The Degrees of Categorical Theories with Recursive Models
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9939(XX)0000-0
THE DEGREES OF CATEGORICAL THEORIES WITH RECURSIVE MODELS URI ANDREWS (Communicated by Julia Knight)
Abstract. We show that even for categorical theories, recursiveness of the models guarantees no information regarding the complexity of the theory. In particular, we show that every tt-degree reducible to 0(ω) contains both ℵ1 categorical and ℵ0 -categorical theories in finite languages all of whose countable models have recursive presentations.
1. Introduction A fundamental question of recursive model theory is to understand the relationship between the complexity of a theory and the complexity of presentations of its models. It is well known that if a theory is recursive, then the Henkin construction produces a decidable model, that is a model whose elementary diagram is recursive. Also, if T is recursive and ℵ1 -categorical, then all of its countable models are decidably presentable [7][10]. On the other hand, if T has a recursive model, that is a model whose atomic diagram is recursive, then in general we can only say that T is tt-reducible to 0(ω) . Naturally, one would like to know whether this bound can be improved upon for tame theories. Two natural classes of tame theories are the ℵ0 -categorical and ℵ1 -categorical theories. Goncharov and Khoussainov [5] showed that for each n there is an ℵ1 -categorical, non-ℵ0 -categorical theory turing equivalent to 0(n) all of whose countable models have recursive presentations. They also showed that for each n there is an ℵ0 categorical theory turing equivalent to 0(n) with a recursive countable model. Fokina [4] extended these result from 0(n) to any arithmetical turing degree. Goncharov and Khoussainov conclude by asking whether there is a theory turing equivalent to 0(ω) which is ℵ1 -categorical and all of its countable models are recursive, and also whether there is an ℵ0 -categorical theory turing equivalent to 0(ω) with a recursive countable model. The latter question was answered by Khoussainov and Montalban [11] in the affirmative. They generalized the construction of the random graph to allow the theory to code true arithmetic. We will answer the former question also in the affirmative. To answer the question, we construct an ℵ1 -categorical, in fact strongly minimal, theory in a finite language with no level of quantifier elimination even after any collection of parameters are named. This cannot be achieved with a disintegrated theory [6]. In fact, any disintegrated theory with a recursive model is 2010 Mathematics Subject Classification. 03C98, 03D99. c
XXXX American Mathematical Society
1
2
URI ANDREWS
turing-below 000 . This bound also holds for modular strongly minimal expansions of groups [2], and trivially for field-like expansions of algebraically closed fields, which are all definitional expansions [14]. Thus the result cannot be achieved with the canonical examples of strongly minimal theories satisfying the Zilber trichotomy. Due to this, we construct the theory via a Hrushovski construction. In fact, we will show the stronger result that if d is any tt-degree reducible to 0(ω) , then there exists a strongly minimal theory T ∈ d in a finite language all of whose models are (uniformly) recursively presentable. There will be two components to the proof of this recursive model theoretic result. One, the infinite language version of the Hrushovski amalgamation construction [1] as presented in section 1.1, is a model theoretic tool for building strongly minimal theories. The other, the Ash-Knight metatheorem [3] as presented in section 1.2, is the recursion theoretic tool to manage 0(ω) -level information. First we will define the theory by the model theoretic construction, and then we will verify by use of the metatheorem that all of the countable models of the theory have recursive presentations. In Section 4, we will similarly show that if d is any tt-degree below 0(ω) , then there exists an ℵ0 -categorical theory T ∈ d in a finite language whose countable model is recursively presentable. The proof follows very similar lines, where we replace the Hrushovski amalgamation by a Fra¨ıss´e amalgamation and we replace the metatheorem argument with a theorem of Knight. 1.1. Infinite language Hrushovski constructions. This section is a summary of sections 2 and 3 of [1], which in turn is an adaptation of Hrushovski’s original construction [9] to an infinite language. There is no new content in this section, and it is included for self-containment of this paper. The key concept of a Hrushovski construction is to fix a function δ on finite L-structures which will serve as an approximation to their dimension. The true dimension of any finite set A will be d(A) = min{δ(B)|B ⊇ A}. We will then form a model by amalgamating finite L-structures as much as possible while ensuring that no finite set has negative dimension and that any extension of dimension 0 has at most a fixed finite number of copies. Let L be a countable relational language. For ease of notation, we assume L contains only ternary relation symbols, say with signature {Ri |i ∈ I}. In the structure we generate, we ensure that each relation symbol is symmetric and holds only on distinct tuples. For a finite L-structure A, we write |R(A)| for the number of tuples from A on which R holds. The following definitions are standard to Hrushovski constructions, and we will use them throughout. Definition 1. For any finite L-structures A and B and infinite L-structure D, we define: P • δ(A) = |A| − i∈I |Ri (A)|. • δ(B/A) = δ(A ∪ B) − δ(A). • If A ⊆ B, we set δ(A, B) = min{δ(C)|A ⊆ C ⊆ B}. • Similarly, if A ⊆ D, we set δ(A, D) = min{δ(C)|A ⊆ C ⊂ D, C finite}. • If A ⊆ B, we say A is strong in B or A ≤ B if δ(A) = δ(A, B). We say A is strong in D if A ⊆ D and A is strong in C for each finite A ⊆ C ⊂ D.
THE DEGREES OF CATEGORICAL THEORIES WITH RECURSIVE MODELS
3
• We say B is simply algebraic over A if A ∩ B = ∅, A ≤ A ∪ B, δ(B/A) = 0, and there is no proper subset B 0 of B such that δ(B 0 /A) = 0. • We say that B is minimally simply algebraic over A if B is simply algebraic over A and there is no proper subset A0 of A such that B is simply algebraic over A0 . • We say A and B are freely joined if δ(A ∪ B) = δ(A) + δ(B) − δ(A ∩ B). Note that A and B are freely joined if and only if all relations on A ∪ B are on tuples entirely from A or entirely from B. The following are key lemmas which verify that δ and ≤ act as expected. Lemma 2. (Submodularity Property) For any finite L-structures A, B ⊆ C, δ(A ∪ B) ≤ δ(A) + δ(B) − δ(A ∩ B). Lemma 3. Let A be a finite L-substructure of N . Suppose A ≤ N . (1) δ(X ∩ A) ≤ δ(X) whenever X ⊆ N is finite. (2) δ(A0 , A) = δ(A0 , N ) whenever A0 ⊆ A. (3) In particular, if A0 ≤ A ≤ N , then A0 ≤ N Lemma 4. If X, A, and B are finite L-structures such that A ⊆ B, then δ(X/A ∪ (X ∩ B)) ≥ δ(X/B). In particular, if X ∩ B = ∅, then δ(X/A) ≥ δ(X/B). Up to this point, the definitions and lemmas have all been as in [9]. The following definition is necessary to allow for the infinite language. We need the condition that we bound the number of extensions of relative dimension 0 to be first order. Since the language is infinite, it would not be first order to bound the number of extensions Y ⊃ X isomorphic to a particular pair B ⊃ A. Instead, we bound the number of extensions Y over X enough like B ⊃ A that the relations showing δ(B/A) = 0 suffice to witness that δ(Y /X) ≤ 0. This idea is the content of the next definition. Definition 5. • For any disjoint L-structures A and B, we write tpr.q.f. (B/A) for the set {Ri (¯ x)|¯ x ⊆ (B ∪ A)3 r A3 , i ∈ I, and Ri (¯ x) holds}. We call this set the relative quantifier-free type of B over A. • Let LB/A be the language generated by {Ri |∃¯ x ∈ (B ∪ A)3 r A3 (Ri (¯ x))}, i.e., the language appearing in tpr.q.f. (B/A). • Suppose Y and X are finite L-structures such that Y is minimally simply algebraic over X and that B and A are finite L-structures such that tpr.q.f. (B/A)|LY /X = tpr.q.f. (Y /X) and tpq.f. (A) = tpq.f. (X). In other words, A is isomorphic to X and up to the language LY /X , B/A is isomorphic to Y /X. Then we say the extension B over A is of the form of Y over X. We will define the amalgamation class with the two conditions that all finite sets have non-negative dimension and that a uniform bound holds on the number of extensions over a set A of the form of Y over X. To do so, we fix a bound function µ with the following properties: • µ is a function from pairs of L-structures to ω such that µ(Y, X) depends only on the atomic type of the pair (Y, X). • For any Y and X, µ(Y, X) ≥ |X|. • If Γ is a relative quantifier-free type, there exists a sublanguage L0 of L with finite subsignature such that whenever tpr.q.f. (Y /X) = Γ = tpr.q.f. (Y 0 /X 0 ) and tpq.f. (X)|L0 = tpq.f. (X 0 )|L0 , then µ(Y, X) = µ(Y 0 , X 0 ).
4
URI ANDREWS
The first condition is necessary for µ to be well defined, the second condition is necessary for the combinatorics of lemma 8, and the last condition is again necessary to ensure that µ is a first-order bound. Definition 6. Let C = Cµ be the class of finite L-structures C such that the following hold: • If A ⊆ C then δ(A) ≥ 0. • Let Y /X be a minimally simply algebraic extension. Let B1 , . . . , Bn , A be disjoint subsets of C such that Bi /A is of the form of Y /X for each i. Then n ≤ µ(Y, X). To generate a model from C, we verify that C has a strong amalgamation property. To see this, some combinatorics must be done. The key to the combinatorics is the following lemma, which will be used again in section 3. Lemma 7. Let A, B1 , B2 be L-structures such that any substructure has nonnegative δ-value, A = B1 ∩ B2 , and A ≤ B1 . Let E be the free-join of B1 with B2 over A. Suppose C 1 , . . . C r , F are disjoint substructures of E such that each C i is minimally simply algebraic over F and the structures C i and C j are isomorphic over F for each 1 ≤ i, j ≤ r. Then one of the following holds: i (1) One of the C in B1 r A and Sr is contained Sr F ⊆ A. (2) Either F ∪ i=1 C i ⊆ B2 or both F ∪ i=1 C i ⊆ B1 and one of the C i is contained in B1 r A. (3) r ≤ δ(F ) (4) For one C j , setting X = (F ∩ A) ∪ (C j ∩ B2 ), δ(X/X ∩ A) < 0. Further, one of the C j is contained in B1 r A. (Note that this cannot happen if A ≤ B2 by Lemma 3). Proof. A careful reading of Lemma 3 of [9] will show that this is indeed what is proved there. Lemma 8. (Strong Amalgamation Lemma) Suppose A, B, C ∈ C, A ≤ B, A ≤ C. Then there exists D ∈ C such that C ≤ D, and an embedding g : B → D such that g(B) ≤ D and g(A) = id|A . Proof. We refer the reader to Lemma 13 of [1].
The following theorem summarizes the results of section 3 of [1]. Theorem 9. Amalgamation of C yields a unique countable generic amalgam M. This M is saturated and strongly minimal. For a ∈ M and finite B ⊂ M, a ∈ acl(B) if and only if d(aB) = d(B). Thus the algebraic dimension of a finite B ⊂ M is d(B). (Recall that d(X) = min{δ(Y )|Y ⊇ X, Y finite}.) 1.2. The Ash-Knight metatheorem. This section entirely follows Ash-Knight [3] (see pg. 236), with the notational exception that the set which we call V is there referred to as L. To maintain as much of the Ash-Knight notation as possible, we still refer to elements of V as l’s. Let V and U be recursively enumerable sets, E be a partial recursive enumeration function on V , and let P be a recursively enumerable alternating tree on V and U made up of non-empty finite sequences which all start with the same ˆl ∈ V . Let (≤n )n∈ω be uniformly recursively enumerable binary relations on V . We define the tuple (V, U, ˆl, P, E, (≤n )n n0 > . . . > nt , then there exists l∗ such that τ ul∗ ∈ P and li ≤ni l∗ for each 0 ≤ i ≤ t.
Theorem 10. (Ash-Knight Metatheorem, [3] Thm 14.4 with α = ω) Let (V, U, ˆl, P, E, (≤n )n µ(Y, X). Restricting E to the language LY /X , we see each of the C j are minimally simply algebraic over F in the same way. Here we have the same set-up as in Hrushovski’s algebraic amalgamation lemma ([9], Lemma 3). Claims 1-3 and case 1 of Hrushovski’s algebraic amalgamation lemma hold exactly as proved there. Case 1 leads to the exception in this lemma. In case 2, we need only count the number of C j which are entirely contained in B1 r A. There are certainly fewer 0 than 2|(B1 rA)∪A | = 2k such C j . Thus, n ≤ |F | + 2k ≤ µ(Y, X). We will write Γi (¯ y, x ¯) to denote the first order formula designating that y¯ over x ¯ is a Γi -extension. Note that the formula Γi (¯ y, x ¯) involves only the relation symbol Ri−1 . k
Lemma 13. For every i ≥ 1, M |= Ri (¯ x) ↔ ¬∃5+2 y¯(Γi (¯ y, x ¯)). (We write m ∃ y¯φ(¯ y ) to represent the formula stating that there exists m disjoint tuples y¯ satisfying φ.) Proof. The rightward direction follows from the fact that any finite substructure of M is an element of C. If the rightward direction did not hold, then we would be violating the µ-bound for Γi -extensions. The leftward direction follows from the previous lemma. Suppose ¬Ri (¯ x) holds. Let B be such that x ¯ ⊆ B ≤ M. Repeated application of the previous lemma amalgamating a Γi -extension with B over x ¯ shows that there is a C ∈ C such that B ≤ C and C contains 5 + 2k disjoint Γi -extensions over x ¯. The fullness of the amalgamation of M (i.e., whenever B ≤ M and B ≤ C, there is an embedding f : C → M so that f (C) ≤ M and f |B = idB . This is property 3 in [1].) guarantees that this C embeds in M over B. Since the first order formula Γi (¯ y, x ¯) is defined using only the relation Ri−1 , we see that each of the Ri are definable via the relation R0 . Definition 14. Let T = T + |R0 . By Lemma 13, T + is a definitional expansion of T . This T is going to be the theory referenced in our main theorem. Note that the defining formulae for Ri from R0 do not depend on S and are uniformly recursive, so T ≡tt T + . The following lemma shows that T + , and thus T , tt-computes S. k
Lemma 15. For every i ≥ 1, i ∈ S if and only if M |= ∀¯ x∃4+2 y¯(Γi (¯ y, x ¯)). Proof. There are as many disjoint Γi -extensions in M over x ¯ as µ allows, by the same argument as in Lemma 13. If i ∈ S, then µ always allows at least 4 + 2k
THE DEGREES OF CATEGORICAL THEORIES WITH RECURSIVE MODELS
7
extensions. If i ∈ / S and Ri (¯ x), then µ allows only 3 + 2k Γi -extensions over x ¯, 4+2k witnessing that M |= ¬∀¯ x∃ y¯(Γi (¯ y, x ¯)). This shows that S ≤tt T + ≡tt T . T is also recursive in S (Properties 2 and 300 from [1] are explicit axiom schemata for T which are recursive in C, which in turn is recursive in S), and since the construction works for any S, this computation is total, yielding S ≡tt T . All that remains is to show that each countable model of T is recursive. 3. Constructing the countable models of T Before using the metatheorem to recursively construct models of T , we recall the construction of a model of T ignoring the recursion theoretic issues. The following will construct a k-dimensional model of T + , i.e., a k-dimensional model of T . Stage 0: Let M0 be the structure with k elements and no relations holding between them. Stage s>0: We are given Ms−1 , the structure constructed at the previous stage. List off the first s minimally simply algebraic extensions over subsets of Ms−1 : B1 /A1 , . . . , Bs /As . If the free amalgam of B1 with Ms−1 over A1 is in C, replace Ms−1 with this amalgam. Repeating this procedure for j ≤ s gives a new structure, which we call Ms . This gives a chain of structures M0 ≤ M1 ≤ . . . Mk . . . ≤ ∪i Mi , where ∪i Mi is the k-dimensional model of T + . We wish to employ this construction, but we cannot recursively compute µ, so we cannot recursively tell whether an amalgam is in C. In fact, the k-dimensional model of T + will not be recursive. We will rather work with approximations to S, thus to µ and C, and will build approximations to a model of T + , but we will injure our Ri -assignments when our approximation to the question of whether i ∈ S changes. Of course, we will use the metatheorem to do this coherently for all i. Recall that whether i ∈ S is uniformly answered by 0(i) . We use the Ash-Knight Metatheorem to construct the k-dimensional model of T by defining an ω-system (V, U, ˆl, P, E, (≤n )n∈ω ). Throughout the construction, we will be working with various estimates to the set S. These estimates will be represented by elements of 2 nt . Without loss of generality, we assume that n0 = n − 1. We need to show that there exists an l∗ such that τ ul∗ ∈ P , and for each i, li ≤ni l∗ . First we will define an auxiliary structure l# = (N , σ) which will handle the injury occurring in this sequence of l’s. To form N , we keep the occurrences of relation symbols from the Mlj to which we are committed, and forget all of the others. Some combinatorics will be required to verify that N ∈ Cu . We will then extend l# to an l∗ which has the right dimension and contains amalgamations of the required minimally simply algebraic extensions from Cu . To avoid notation such as Mlj , we write lj = (Mj , σj ). Let l# be the pair (N , σ), defined as follows: σ = u and N has the same universe as Mt . Let x ¯ be a tuple in N . We now describe whether or not Ri holds on x ¯. Let m be least such that x ¯ ⊆ Mm . Then Ri holds on x ¯ in N if and only if Ri is in Lnm and holds on x ¯ in Mm .
THE DEGREES OF CATEGORICAL THEORIES WITH RECURSIVE MODELS
9
We will write Ni for the substructure of N with the same universe as Mi , and we will write Li for Lni . Claim 19. Mi |Li = Ni |Li , i.e., for each relation in Li and every tuple in Mi , the relation holds in Mi if and only if it holds in N . Proof. For each j ≤ i, lj ≤ni−1 li . In particular, lj ≤ni li . Let x ¯ be any tuple in Mi and let m be minimal such that x ¯ ⊆ Mm . Then m ≤ i, so lm ≤ni li . Thus for R ∈ Li , R(¯ x) holds in Mm if and only if it holds in N (by definition of N ), and the first condition is equivalent to R(¯ x) holding in Mi , since lm ≤ni li . In particular, N0 = M0 since n0 = n − 1. Lemma 20. l# ∈ V Proof. We need to verify that N ∈ Cσ . We verify this by verifying each condition in the definition of Cσ . (1) δ(A) ≥ 0 for all A ⊆ N Proof. Let A be a subset of N . Let Ai = A ∩ Ni . We need to show that δ(At ) ≥ 0. We achieve this by showing that δ(Ai+1 /Ai ) ≥ 0 for each i. This suffices since δ(At ) = δ(At /At−1 ) + δ(At−1 /At−2 ) + . . . + δ(A1 /A0 ) + δ(A0 ) and A0 is a subset of N0 = M0 , hence has non-negative δ-value. δ(Ai+1 /Ai ) is |Ai+1 r Ai | − (the number of relations holding in Ai+1 involving at least one element in Ai+1 rAi ). Consider B the subset of Mi+1 with the same underlying set as Ai+1 . Since Mi |Li ≤ Mi+1 |Li , δ(B|Li /(B ∩ Mi )|Li ) ≥ 0, but δ(B|Li /(B ∩ Mi )|Li ) ≤ δ(Ai+1 /Ai ) as every relation counting on the right counts on the left as well. Thus, each summand is non-negative and δ(A) ≥ 0. (2) If C 1 , . . . , C n , F are disjoint subsets of N , and each C j over F is of the form of Y over X, a minimally simply algebraic extension, then n ≤ µσ (Y, X). Proof. We proceed by induction to show that the condition holds for each Ni . The condition holds on N0 , as this is just M0 and µσ agrees with µσ0 . Supposing the condition holds for Ns−1 , we will show that the condition holds on Ns as well. The proof follows via Lemma 7. Suppose C 1 , . . . , C n , F are disjoint subsets of Ns , and each C j over F is of the form of Y over X. Since Ns−1 ≤ Ns , we apply Lemma 7 with B1 = Ns |LY /X , A = B2 = Ns−1 |LY /X . There are 4 cases to consider. In one case, r ≤ |X| < µσ (Y, X). In each of the other cases, one C j is entirely contained in Ns r Ns−1 . In the case that one C j is entirely contained in Ns r Ns−1 , tpr.q.f. (Y /X) only includes relations from the language Ls . There are a number of possibilities to consider: case 1: Y over X is not a Γi -extension for any i. Let F0 be the subset of Ms with underlying set the same as F . Since Ms |Ls = Ns |Ls , we see that each of the C j over F , looked at as subsets of Ms , are of the form of Y over X 0 , where X 0 ∼ = F0 . Since for nonΓi -extensions, µσ (Y, X) only depends on |X|, and since Ms satisfies the property for µσs , we have n ≤ µσs (Y, X 0 ) = µσ (Y, X). case 2: Y over X is a Γi -extension and ¬Ri (X).
10
URI ANDREWS
In this case, we also look at the C j and F as subsets of Ms . The C j are each Γi -extensions over F . If Ri (F ) is in Ms , then the number of C j is bounded by µσs (Y, X 0 ) (where X 0 is the same as X but with Ri (X) holding) which is even less than µσs (Y, X). If ¬Ri (F ) in Ms , then the number of C j is bounded by µσs (Y, X). Since σs |ns−1 = σ|ns−1 , µσs (Y, X) = µσ (Y, X). case 3: Y over X is a Γi -extension and Ri (X). If Ri ∈ Ls , then we have the corresponding fact in Ms , so we get the µ-bound from the fact that Ms satisfies the property for µσs and σs |ns−1 = σ|ns−1 . Now suppose Ri ∈ / Ls . Since Y over X is a Γi -extension, Ri−1 ∈ Ls . So Ri ∈ Ls−1 . Clearly, Ri (F ) implies that F ⊆ Ns−1 . But then Ri (F ) holds in Ms−1 , and ls−1 ≤ns−1 ls , so Ri (F ) holds in Ms as well. Again, we get the µ-bound from Ms . This concludes the inductive step, showing that N = Nt satisfies the condition. Claim 21. For each i, li ≤ni l# . Proof. We verify the two properties. First we verify σi |ni = σ|ni . We know that l0 ≤ni−1 li , so σ|ni = σ0 |ni = σi |ni . Second we verify that Mi |Li ≤ N |Li . Claim 19 gives us that Mi |Li ⊂ N |Li . Now, let X be a subset of N |Li . We use the same argument as before (when we showed that δ(A) ≥ 0 for all A ⊆ N ). We need to show that δ(X/Ni |Li ) ≥ 0. For i ≤ j ≤ t, we write Xj = ((X ∩ Nj ) ∪ Ni )|Li . Then we have δ(X/Ni |Li ) = δ(Xt /Xt−1 ) + δ(Xt−1 /Xt−2 ) + . . . + δ(Xi+1 /Xi ). As in the previous argument, each summand is non-negative, so δ(X/Ni |Li ) ≥ 0. The only obstructions to l# being what we need for l∗ is that it might not contain enough copies of the first n minimally simply algebraic extensions, and perhaps δ(N ) > k. Extend N using only the thus far unused relation symbol Rn−1 to N 0 so that δ(N 0 ) = k, M0 ≤ N 0 , and N 0 ∈ Cσ . Then proceed to extend N 0 to N ∗ by amalgamating as many copies as allowed in Cσ of the first n minimally simply algebraic extensions over N ∗ . We set l∗ to be (N ∗ , u). By construction, li ≤ni l# ≤ni l∗ and τ ul∗ ∈ P as l∗ ∈ V , M0 ≤ N ∗ , σl∗ = u, and δ(N ∗ ) = k. Having found this l∗ , we have shown that (V, U, ˆl, E, P, (≤n )n∈ω ) is an ω-system. We have a uniformly 0(n) instruction function for u, namely un = S|n , the string in 2 3: If i − 10 ∈ / S or i < 10, then
^
M |= Ri (¯ x) ↔ ¬∃y, z P (y, z) ∧
(Ri−1 (y, w) ¯ ∧ Ri−1 (z, w)) ¯
w⊂¯ ¯ x,|w|=i−2 ¯
Similarly, if i − 10 ∈ S, then
^
M |= Ri (¯ x) ↔ ¬∃y, z Q(y, z) ∧
(Ri−1 (y, w) ¯ ∧ Ri−1 (z, w)) ¯
w⊂¯ ¯ x,|w|=i−2 ¯
Proof. The rightward direction follows via the fact that Age(M ) = K. To show the leftward direction, take a tuple x ¯ such that M |= ¬Ri (¯ x). By ultrahomogeneity of M, it suffices to show that x ¯ embeds into an element of K where such a y and z exist. Consider the structure A = x ¯ ∪ {a, b}, where each of a and b are Ri−1 -related to every (i − 2)-element subset of x ¯, a and b are P -related (Qrelated in the case of the second equivalence above), and no other relations hold involving a or b. It is easy to verify that this structure is in K. Definition 29. Let T = Th(M)|{P,Q,R3 } By the previous lemma, Th(M) is a definitional expansion of T . Further, from T we can in a tt way determine which definition of Ri is correct, and thus whether i − 10 ∈ S. This shows that T ≥tt S. Further, by ultra-homogeneity of T , T ≤tt K ≤tt S. It remains only to prove that the countable model of T is recursively presentable. Lemma 30. Uniformly in n, T ∩ ∃n is computable in 0n−7 . Proof. We first show that any ∃n formula in T is equivalent to a quantifier-free formula in the relations {P, Q, Ri }i≤3+n . The proof proceeds by setting up the appropriate Ehrenfeucht-Fra¨ıss´e game (see: [13], section 2.4) and seeing that ‘∃loise’ (the second player) has a winning strategy. The game is the standard EhrenfeuchtFra¨ıss´e game of length n where we start with tuples a ¯ and ¯b which have the same {P, Q, Ri }i≤3+n -quantifier-free type. Then whichever tuple c¯ ‘∀belard’ chooses, ∃loise can choose a tuple d¯ so that a ¯c¯ and ¯bd¯ satisfy the same {P, Q, Ri }i≤3+n−1 quantifier-free type. Proceeding as such, ∃loise wins the game of length n. This shows that any ∃n formula depends only on the relations {P, Q, Ri }i≤3+n ([13], Lemma 2.4.9). Thus, the ∃n formula ∃∀ . . . φ(¯ x) is equivalent to: _ ^ . . . φ(¯ x) (configurations in {P,Q,Ri }i≤3+n in K) (configurations in {P,Q,Ri }i≤3+n−1 in K)
THE DEGREES OF CATEGORICAL THEORIES WITH RECURSIVE MODELS
13
To verify whether this statement is true, we need only to be able to parse “in K” for configurations in the language {P, Q, Ri }i≤3+n . The conditions of being in K is then described recursively in 03+n−10 . We use the following case of a theorem of Knight [12] to show that the countable model of T is recursively presentable. Theorem 31. (Knight) Let T be an ℵ0 -categorical theory. If T ∩ ∃n+1 is Σ0n uniformly in n, then T has a recursive model. Since T is ℵ0 -categorical, Lemma 30 shows that T satisfies the conditions of this theorem. Thus we conclude the promised theorem. Theorem 32. Let d be a tt-degree reducible to 0(ω) . Then there exists an ℵ0 categorical theory T ∈ d in a finite language whose countable model is recursively presentable. Proof. As above, d contains a set S such that n ∈ S is uniformly recursive in 0(n) . Applying the construction to this set S yields an ℵ0 -categorical theory T ∈ d in a finite language whose countable model is recursively presentable. Corollary 33. There exists an ℵ0 -categorical theory T in a finite language whose countable model is recursively presentable and T ≡T 0(ω) . References 1. Uri Andrews, A new spectrum of recursive models using an amalgamation construction, J. Symbolic Logic 76 (2011), 883–896. 2. Uri Andrews and Alice Medvedev, Recursive spectra of strongly minimal theories satisfying the zilber trichotomy, submitted for publication. 3. C. J. Ash and J. F. Knight, Computable structures and the hyperarithmetical hierarchy, Studies in Logic and the Foundations of Mathematics, vol. 144, North-Holland Publishing Co., Amsterdam, 2000. 4. Ekaterina Fokina, Arithmetic Turing degrees and categorical theories of computable models, Mathematical logic in Asia, World Sci. Publ., Hackensack, NJ, 2006, pp. 58–69. 5. Sergei S. Goncharov and Bakhadyr Khoussainov, Complexity of theories of computable categorical models, Algebra Logika 43 (2004), no. 6, 650–665, 758–759. 6. Sergey S. Goncharov, Valentina S. Harizanov, Michael C. Laskowski, Steffen Lempp, and Charles F. D. McCoy, Trivial, strongly minimal theories are model complete after naming constants, Proc. Amer. Math. Soc. 131 (2003), no. 12, 3901–3912 (electronic). 7. Leo Harrington, Recursively presentable prime models, J. Symbolic Logic 39 (1974), 305–309. 8. Wilfrid Hodges, A shorter model theory, Cambridge University Press, Cambridge, 1997. 9. Ehud Hrushovski, A new strongly minimal set, Ann. Pure Appl. Logic 62 (1993), no. 2, 147–166, Stability in model theory, III (Trento, 1991). 10. Nazif G. Khisamiev, Strongly constructive models of a decidable theory, Izv. Akad. Nauk Kazah. SSR Ser. Fiz.-Mat. (1974), no. 1, 83–84, 94. 11. Bakhadyr Khoussainov and Antonio Montalb´ an, A computable ℵ0 -categorical structure whose theory computes true arithmetic, J. Symbolic Logic 75 (2010), no. 2, 728–740. 12. Julia F. Knight, Nonarithmetical ℵ0 -categorical theories with recursive models, J. Symbolic Logic 59 (1994), no. 1, 106–112. 13. David Marker, Model theory, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, New York, 2002, An introduction. 14. Bruno Poizat, MM. Borel, Tits, Zil0 ber et le G´ en´ eral Nonsense, J. Symbolic Logic 53 (1988), no. 1, 124–131.
THE DEGREES OF CATEGORICAL THEORIES WITH RECURSIVE MODELS URI ANDREWS (Communicated by Julia Knight)
Abstract. We show that even for categorical theories, recursiveness of the models guarantees no information regarding the complexity of the theory. In particular, we show that every tt-degree reducible to 0(ω) contains both ℵ1 categorical and ℵ0 -categorical theories in finite languages all of whose countable models have recursive presentations.
1. Introduction A fundamental question of recursive model theory is to understand the relationship between the complexity of a theory and the complexity of presentations of its models. It is well known that if a theory is recursive, then the Henkin construction produces a decidable model, that is a model whose elementary diagram is recursive. Also, if T is recursive and ℵ1 -categorical, then all of its countable models are decidably presentable [7][10]. On the other hand, if T has a recursive model, that is a model whose atomic diagram is recursive, then in general we can only say that T is tt-reducible to 0(ω) . Naturally, one would like to know whether this bound can be improved upon for tame theories. Two natural classes of tame theories are the ℵ0 -categorical and ℵ1 -categorical theories. Goncharov and Khoussainov [5] showed that for each n there is an ℵ1 -categorical, non-ℵ0 -categorical theory turing equivalent to 0(n) all of whose countable models have recursive presentations. They also showed that for each n there is an ℵ0 categorical theory turing equivalent to 0(n) with a recursive countable model. Fokina [4] extended these result from 0(n) to any arithmetical turing degree. Goncharov and Khoussainov conclude by asking whether there is a theory turing equivalent to 0(ω) which is ℵ1 -categorical and all of its countable models are recursive, and also whether there is an ℵ0 -categorical theory turing equivalent to 0(ω) with a recursive countable model. The latter question was answered by Khoussainov and Montalban [11] in the affirmative. They generalized the construction of the random graph to allow the theory to code true arithmetic. We will answer the former question also in the affirmative. To answer the question, we construct an ℵ1 -categorical, in fact strongly minimal, theory in a finite language with no level of quantifier elimination even after any collection of parameters are named. This cannot be achieved with a disintegrated theory [6]. In fact, any disintegrated theory with a recursive model is 2010 Mathematics Subject Classification. 03C98, 03D99. c
XXXX American Mathematical Society
1
2
URI ANDREWS
turing-below 000 . This bound also holds for modular strongly minimal expansions of groups [2], and trivially for field-like expansions of algebraically closed fields, which are all definitional expansions [14]. Thus the result cannot be achieved with the canonical examples of strongly minimal theories satisfying the Zilber trichotomy. Due to this, we construct the theory via a Hrushovski construction. In fact, we will show the stronger result that if d is any tt-degree reducible to 0(ω) , then there exists a strongly minimal theory T ∈ d in a finite language all of whose models are (uniformly) recursively presentable. There will be two components to the proof of this recursive model theoretic result. One, the infinite language version of the Hrushovski amalgamation construction [1] as presented in section 1.1, is a model theoretic tool for building strongly minimal theories. The other, the Ash-Knight metatheorem [3] as presented in section 1.2, is the recursion theoretic tool to manage 0(ω) -level information. First we will define the theory by the model theoretic construction, and then we will verify by use of the metatheorem that all of the countable models of the theory have recursive presentations. In Section 4, we will similarly show that if d is any tt-degree below 0(ω) , then there exists an ℵ0 -categorical theory T ∈ d in a finite language whose countable model is recursively presentable. The proof follows very similar lines, where we replace the Hrushovski amalgamation by a Fra¨ıss´e amalgamation and we replace the metatheorem argument with a theorem of Knight. 1.1. Infinite language Hrushovski constructions. This section is a summary of sections 2 and 3 of [1], which in turn is an adaptation of Hrushovski’s original construction [9] to an infinite language. There is no new content in this section, and it is included for self-containment of this paper. The key concept of a Hrushovski construction is to fix a function δ on finite L-structures which will serve as an approximation to their dimension. The true dimension of any finite set A will be d(A) = min{δ(B)|B ⊇ A}. We will then form a model by amalgamating finite L-structures as much as possible while ensuring that no finite set has negative dimension and that any extension of dimension 0 has at most a fixed finite number of copies. Let L be a countable relational language. For ease of notation, we assume L contains only ternary relation symbols, say with signature {Ri |i ∈ I}. In the structure we generate, we ensure that each relation symbol is symmetric and holds only on distinct tuples. For a finite L-structure A, we write |R(A)| for the number of tuples from A on which R holds. The following definitions are standard to Hrushovski constructions, and we will use them throughout. Definition 1. For any finite L-structures A and B and infinite L-structure D, we define: P • δ(A) = |A| − i∈I |Ri (A)|. • δ(B/A) = δ(A ∪ B) − δ(A). • If A ⊆ B, we set δ(A, B) = min{δ(C)|A ⊆ C ⊆ B}. • Similarly, if A ⊆ D, we set δ(A, D) = min{δ(C)|A ⊆ C ⊂ D, C finite}. • If A ⊆ B, we say A is strong in B or A ≤ B if δ(A) = δ(A, B). We say A is strong in D if A ⊆ D and A is strong in C for each finite A ⊆ C ⊂ D.
THE DEGREES OF CATEGORICAL THEORIES WITH RECURSIVE MODELS
3
• We say B is simply algebraic over A if A ∩ B = ∅, A ≤ A ∪ B, δ(B/A) = 0, and there is no proper subset B 0 of B such that δ(B 0 /A) = 0. • We say that B is minimally simply algebraic over A if B is simply algebraic over A and there is no proper subset A0 of A such that B is simply algebraic over A0 . • We say A and B are freely joined if δ(A ∪ B) = δ(A) + δ(B) − δ(A ∩ B). Note that A and B are freely joined if and only if all relations on A ∪ B are on tuples entirely from A or entirely from B. The following are key lemmas which verify that δ and ≤ act as expected. Lemma 2. (Submodularity Property) For any finite L-structures A, B ⊆ C, δ(A ∪ B) ≤ δ(A) + δ(B) − δ(A ∩ B). Lemma 3. Let A be a finite L-substructure of N . Suppose A ≤ N . (1) δ(X ∩ A) ≤ δ(X) whenever X ⊆ N is finite. (2) δ(A0 , A) = δ(A0 , N ) whenever A0 ⊆ A. (3) In particular, if A0 ≤ A ≤ N , then A0 ≤ N Lemma 4. If X, A, and B are finite L-structures such that A ⊆ B, then δ(X/A ∪ (X ∩ B)) ≥ δ(X/B). In particular, if X ∩ B = ∅, then δ(X/A) ≥ δ(X/B). Up to this point, the definitions and lemmas have all been as in [9]. The following definition is necessary to allow for the infinite language. We need the condition that we bound the number of extensions of relative dimension 0 to be first order. Since the language is infinite, it would not be first order to bound the number of extensions Y ⊃ X isomorphic to a particular pair B ⊃ A. Instead, we bound the number of extensions Y over X enough like B ⊃ A that the relations showing δ(B/A) = 0 suffice to witness that δ(Y /X) ≤ 0. This idea is the content of the next definition. Definition 5. • For any disjoint L-structures A and B, we write tpr.q.f. (B/A) for the set {Ri (¯ x)|¯ x ⊆ (B ∪ A)3 r A3 , i ∈ I, and Ri (¯ x) holds}. We call this set the relative quantifier-free type of B over A. • Let LB/A be the language generated by {Ri |∃¯ x ∈ (B ∪ A)3 r A3 (Ri (¯ x))}, i.e., the language appearing in tpr.q.f. (B/A). • Suppose Y and X are finite L-structures such that Y is minimally simply algebraic over X and that B and A are finite L-structures such that tpr.q.f. (B/A)|LY /X = tpr.q.f. (Y /X) and tpq.f. (A) = tpq.f. (X). In other words, A is isomorphic to X and up to the language LY /X , B/A is isomorphic to Y /X. Then we say the extension B over A is of the form of Y over X. We will define the amalgamation class with the two conditions that all finite sets have non-negative dimension and that a uniform bound holds on the number of extensions over a set A of the form of Y over X. To do so, we fix a bound function µ with the following properties: • µ is a function from pairs of L-structures to ω such that µ(Y, X) depends only on the atomic type of the pair (Y, X). • For any Y and X, µ(Y, X) ≥ |X|. • If Γ is a relative quantifier-free type, there exists a sublanguage L0 of L with finite subsignature such that whenever tpr.q.f. (Y /X) = Γ = tpr.q.f. (Y 0 /X 0 ) and tpq.f. (X)|L0 = tpq.f. (X 0 )|L0 , then µ(Y, X) = µ(Y 0 , X 0 ).
4
URI ANDREWS
The first condition is necessary for µ to be well defined, the second condition is necessary for the combinatorics of lemma 8, and the last condition is again necessary to ensure that µ is a first-order bound. Definition 6. Let C = Cµ be the class of finite L-structures C such that the following hold: • If A ⊆ C then δ(A) ≥ 0. • Let Y /X be a minimally simply algebraic extension. Let B1 , . . . , Bn , A be disjoint subsets of C such that Bi /A is of the form of Y /X for each i. Then n ≤ µ(Y, X). To generate a model from C, we verify that C has a strong amalgamation property. To see this, some combinatorics must be done. The key to the combinatorics is the following lemma, which will be used again in section 3. Lemma 7. Let A, B1 , B2 be L-structures such that any substructure has nonnegative δ-value, A = B1 ∩ B2 , and A ≤ B1 . Let E be the free-join of B1 with B2 over A. Suppose C 1 , . . . C r , F are disjoint substructures of E such that each C i is minimally simply algebraic over F and the structures C i and C j are isomorphic over F for each 1 ≤ i, j ≤ r. Then one of the following holds: i (1) One of the C in B1 r A and Sr is contained Sr F ⊆ A. (2) Either F ∪ i=1 C i ⊆ B2 or both F ∪ i=1 C i ⊆ B1 and one of the C i is contained in B1 r A. (3) r ≤ δ(F ) (4) For one C j , setting X = (F ∩ A) ∪ (C j ∩ B2 ), δ(X/X ∩ A) < 0. Further, one of the C j is contained in B1 r A. (Note that this cannot happen if A ≤ B2 by Lemma 3). Proof. A careful reading of Lemma 3 of [9] will show that this is indeed what is proved there. Lemma 8. (Strong Amalgamation Lemma) Suppose A, B, C ∈ C, A ≤ B, A ≤ C. Then there exists D ∈ C such that C ≤ D, and an embedding g : B → D such that g(B) ≤ D and g(A) = id|A . Proof. We refer the reader to Lemma 13 of [1].
The following theorem summarizes the results of section 3 of [1]. Theorem 9. Amalgamation of C yields a unique countable generic amalgam M. This M is saturated and strongly minimal. For a ∈ M and finite B ⊂ M, a ∈ acl(B) if and only if d(aB) = d(B). Thus the algebraic dimension of a finite B ⊂ M is d(B). (Recall that d(X) = min{δ(Y )|Y ⊇ X, Y finite}.) 1.2. The Ash-Knight metatheorem. This section entirely follows Ash-Knight [3] (see pg. 236), with the notational exception that the set which we call V is there referred to as L. To maintain as much of the Ash-Knight notation as possible, we still refer to elements of V as l’s. Let V and U be recursively enumerable sets, E be a partial recursive enumeration function on V , and let P be a recursively enumerable alternating tree on V and U made up of non-empty finite sequences which all start with the same ˆl ∈ V . Let (≤n )n∈ω be uniformly recursively enumerable binary relations on V . We define the tuple (V, U, ˆl, P, E, (≤n )n n0 > . . . > nt , then there exists l∗ such that τ ul∗ ∈ P and li ≤ni l∗ for each 0 ≤ i ≤ t.
Theorem 10. (Ash-Knight Metatheorem, [3] Thm 14.4 with α = ω) Let (V, U, ˆl, P, E, (≤n )n µ(Y, X). Restricting E to the language LY /X , we see each of the C j are minimally simply algebraic over F in the same way. Here we have the same set-up as in Hrushovski’s algebraic amalgamation lemma ([9], Lemma 3). Claims 1-3 and case 1 of Hrushovski’s algebraic amalgamation lemma hold exactly as proved there. Case 1 leads to the exception in this lemma. In case 2, we need only count the number of C j which are entirely contained in B1 r A. There are certainly fewer 0 than 2|(B1 rA)∪A | = 2k such C j . Thus, n ≤ |F | + 2k ≤ µ(Y, X). We will write Γi (¯ y, x ¯) to denote the first order formula designating that y¯ over x ¯ is a Γi -extension. Note that the formula Γi (¯ y, x ¯) involves only the relation symbol Ri−1 . k
Lemma 13. For every i ≥ 1, M |= Ri (¯ x) ↔ ¬∃5+2 y¯(Γi (¯ y, x ¯)). (We write m ∃ y¯φ(¯ y ) to represent the formula stating that there exists m disjoint tuples y¯ satisfying φ.) Proof. The rightward direction follows from the fact that any finite substructure of M is an element of C. If the rightward direction did not hold, then we would be violating the µ-bound for Γi -extensions. The leftward direction follows from the previous lemma. Suppose ¬Ri (¯ x) holds. Let B be such that x ¯ ⊆ B ≤ M. Repeated application of the previous lemma amalgamating a Γi -extension with B over x ¯ shows that there is a C ∈ C such that B ≤ C and C contains 5 + 2k disjoint Γi -extensions over x ¯. The fullness of the amalgamation of M (i.e., whenever B ≤ M and B ≤ C, there is an embedding f : C → M so that f (C) ≤ M and f |B = idB . This is property 3 in [1].) guarantees that this C embeds in M over B. Since the first order formula Γi (¯ y, x ¯) is defined using only the relation Ri−1 , we see that each of the Ri are definable via the relation R0 . Definition 14. Let T = T + |R0 . By Lemma 13, T + is a definitional expansion of T . This T is going to be the theory referenced in our main theorem. Note that the defining formulae for Ri from R0 do not depend on S and are uniformly recursive, so T ≡tt T + . The following lemma shows that T + , and thus T , tt-computes S. k
Lemma 15. For every i ≥ 1, i ∈ S if and only if M |= ∀¯ x∃4+2 y¯(Γi (¯ y, x ¯)). Proof. There are as many disjoint Γi -extensions in M over x ¯ as µ allows, by the same argument as in Lemma 13. If i ∈ S, then µ always allows at least 4 + 2k
THE DEGREES OF CATEGORICAL THEORIES WITH RECURSIVE MODELS
7
extensions. If i ∈ / S and Ri (¯ x), then µ allows only 3 + 2k Γi -extensions over x ¯, 4+2k witnessing that M |= ¬∀¯ x∃ y¯(Γi (¯ y, x ¯)). This shows that S ≤tt T + ≡tt T . T is also recursive in S (Properties 2 and 300 from [1] are explicit axiom schemata for T which are recursive in C, which in turn is recursive in S), and since the construction works for any S, this computation is total, yielding S ≡tt T . All that remains is to show that each countable model of T is recursive. 3. Constructing the countable models of T Before using the metatheorem to recursively construct models of T , we recall the construction of a model of T ignoring the recursion theoretic issues. The following will construct a k-dimensional model of T + , i.e., a k-dimensional model of T . Stage 0: Let M0 be the structure with k elements and no relations holding between them. Stage s>0: We are given Ms−1 , the structure constructed at the previous stage. List off the first s minimally simply algebraic extensions over subsets of Ms−1 : B1 /A1 , . . . , Bs /As . If the free amalgam of B1 with Ms−1 over A1 is in C, replace Ms−1 with this amalgam. Repeating this procedure for j ≤ s gives a new structure, which we call Ms . This gives a chain of structures M0 ≤ M1 ≤ . . . Mk . . . ≤ ∪i Mi , where ∪i Mi is the k-dimensional model of T + . We wish to employ this construction, but we cannot recursively compute µ, so we cannot recursively tell whether an amalgam is in C. In fact, the k-dimensional model of T + will not be recursive. We will rather work with approximations to S, thus to µ and C, and will build approximations to a model of T + , but we will injure our Ri -assignments when our approximation to the question of whether i ∈ S changes. Of course, we will use the metatheorem to do this coherently for all i. Recall that whether i ∈ S is uniformly answered by 0(i) . We use the Ash-Knight Metatheorem to construct the k-dimensional model of T by defining an ω-system (V, U, ˆl, P, E, (≤n )n∈ω ). Throughout the construction, we will be working with various estimates to the set S. These estimates will be represented by elements of 2 nt . Without loss of generality, we assume that n0 = n − 1. We need to show that there exists an l∗ such that τ ul∗ ∈ P , and for each i, li ≤ni l∗ . First we will define an auxiliary structure l# = (N , σ) which will handle the injury occurring in this sequence of l’s. To form N , we keep the occurrences of relation symbols from the Mlj to which we are committed, and forget all of the others. Some combinatorics will be required to verify that N ∈ Cu . We will then extend l# to an l∗ which has the right dimension and contains amalgamations of the required minimally simply algebraic extensions from Cu . To avoid notation such as Mlj , we write lj = (Mj , σj ). Let l# be the pair (N , σ), defined as follows: σ = u and N has the same universe as Mt . Let x ¯ be a tuple in N . We now describe whether or not Ri holds on x ¯. Let m be least such that x ¯ ⊆ Mm . Then Ri holds on x ¯ in N if and only if Ri is in Lnm and holds on x ¯ in Mm .
THE DEGREES OF CATEGORICAL THEORIES WITH RECURSIVE MODELS
9
We will write Ni for the substructure of N with the same universe as Mi , and we will write Li for Lni . Claim 19. Mi |Li = Ni |Li , i.e., for each relation in Li and every tuple in Mi , the relation holds in Mi if and only if it holds in N . Proof. For each j ≤ i, lj ≤ni−1 li . In particular, lj ≤ni li . Let x ¯ be any tuple in Mi and let m be minimal such that x ¯ ⊆ Mm . Then m ≤ i, so lm ≤ni li . Thus for R ∈ Li , R(¯ x) holds in Mm if and only if it holds in N (by definition of N ), and the first condition is equivalent to R(¯ x) holding in Mi , since lm ≤ni li . In particular, N0 = M0 since n0 = n − 1. Lemma 20. l# ∈ V Proof. We need to verify that N ∈ Cσ . We verify this by verifying each condition in the definition of Cσ . (1) δ(A) ≥ 0 for all A ⊆ N Proof. Let A be a subset of N . Let Ai = A ∩ Ni . We need to show that δ(At ) ≥ 0. We achieve this by showing that δ(Ai+1 /Ai ) ≥ 0 for each i. This suffices since δ(At ) = δ(At /At−1 ) + δ(At−1 /At−2 ) + . . . + δ(A1 /A0 ) + δ(A0 ) and A0 is a subset of N0 = M0 , hence has non-negative δ-value. δ(Ai+1 /Ai ) is |Ai+1 r Ai | − (the number of relations holding in Ai+1 involving at least one element in Ai+1 rAi ). Consider B the subset of Mi+1 with the same underlying set as Ai+1 . Since Mi |Li ≤ Mi+1 |Li , δ(B|Li /(B ∩ Mi )|Li ) ≥ 0, but δ(B|Li /(B ∩ Mi )|Li ) ≤ δ(Ai+1 /Ai ) as every relation counting on the right counts on the left as well. Thus, each summand is non-negative and δ(A) ≥ 0. (2) If C 1 , . . . , C n , F are disjoint subsets of N , and each C j over F is of the form of Y over X, a minimally simply algebraic extension, then n ≤ µσ (Y, X). Proof. We proceed by induction to show that the condition holds for each Ni . The condition holds on N0 , as this is just M0 and µσ agrees with µσ0 . Supposing the condition holds for Ns−1 , we will show that the condition holds on Ns as well. The proof follows via Lemma 7. Suppose C 1 , . . . , C n , F are disjoint subsets of Ns , and each C j over F is of the form of Y over X. Since Ns−1 ≤ Ns , we apply Lemma 7 with B1 = Ns |LY /X , A = B2 = Ns−1 |LY /X . There are 4 cases to consider. In one case, r ≤ |X| < µσ (Y, X). In each of the other cases, one C j is entirely contained in Ns r Ns−1 . In the case that one C j is entirely contained in Ns r Ns−1 , tpr.q.f. (Y /X) only includes relations from the language Ls . There are a number of possibilities to consider: case 1: Y over X is not a Γi -extension for any i. Let F0 be the subset of Ms with underlying set the same as F . Since Ms |Ls = Ns |Ls , we see that each of the C j over F , looked at as subsets of Ms , are of the form of Y over X 0 , where X 0 ∼ = F0 . Since for nonΓi -extensions, µσ (Y, X) only depends on |X|, and since Ms satisfies the property for µσs , we have n ≤ µσs (Y, X 0 ) = µσ (Y, X). case 2: Y over X is a Γi -extension and ¬Ri (X).
10
URI ANDREWS
In this case, we also look at the C j and F as subsets of Ms . The C j are each Γi -extensions over F . If Ri (F ) is in Ms , then the number of C j is bounded by µσs (Y, X 0 ) (where X 0 is the same as X but with Ri (X) holding) which is even less than µσs (Y, X). If ¬Ri (F ) in Ms , then the number of C j is bounded by µσs (Y, X). Since σs |ns−1 = σ|ns−1 , µσs (Y, X) = µσ (Y, X). case 3: Y over X is a Γi -extension and Ri (X). If Ri ∈ Ls , then we have the corresponding fact in Ms , so we get the µ-bound from the fact that Ms satisfies the property for µσs and σs |ns−1 = σ|ns−1 . Now suppose Ri ∈ / Ls . Since Y over X is a Γi -extension, Ri−1 ∈ Ls . So Ri ∈ Ls−1 . Clearly, Ri (F ) implies that F ⊆ Ns−1 . But then Ri (F ) holds in Ms−1 , and ls−1 ≤ns−1 ls , so Ri (F ) holds in Ms as well. Again, we get the µ-bound from Ms . This concludes the inductive step, showing that N = Nt satisfies the condition. Claim 21. For each i, li ≤ni l# . Proof. We verify the two properties. First we verify σi |ni = σ|ni . We know that l0 ≤ni−1 li , so σ|ni = σ0 |ni = σi |ni . Second we verify that Mi |Li ≤ N |Li . Claim 19 gives us that Mi |Li ⊂ N |Li . Now, let X be a subset of N |Li . We use the same argument as before (when we showed that δ(A) ≥ 0 for all A ⊆ N ). We need to show that δ(X/Ni |Li ) ≥ 0. For i ≤ j ≤ t, we write Xj = ((X ∩ Nj ) ∪ Ni )|Li . Then we have δ(X/Ni |Li ) = δ(Xt /Xt−1 ) + δ(Xt−1 /Xt−2 ) + . . . + δ(Xi+1 /Xi ). As in the previous argument, each summand is non-negative, so δ(X/Ni |Li ) ≥ 0. The only obstructions to l# being what we need for l∗ is that it might not contain enough copies of the first n minimally simply algebraic extensions, and perhaps δ(N ) > k. Extend N using only the thus far unused relation symbol Rn−1 to N 0 so that δ(N 0 ) = k, M0 ≤ N 0 , and N 0 ∈ Cσ . Then proceed to extend N 0 to N ∗ by amalgamating as many copies as allowed in Cσ of the first n minimally simply algebraic extensions over N ∗ . We set l∗ to be (N ∗ , u). By construction, li ≤ni l# ≤ni l∗ and τ ul∗ ∈ P as l∗ ∈ V , M0 ≤ N ∗ , σl∗ = u, and δ(N ∗ ) = k. Having found this l∗ , we have shown that (V, U, ˆl, E, P, (≤n )n∈ω ) is an ω-system. We have a uniformly 0(n) instruction function for u, namely un = S|n , the string in 2 3: If i − 10 ∈ / S or i < 10, then
^
M |= Ri (¯ x) ↔ ¬∃y, z P (y, z) ∧
(Ri−1 (y, w) ¯ ∧ Ri−1 (z, w)) ¯
w⊂¯ ¯ x,|w|=i−2 ¯
Similarly, if i − 10 ∈ S, then
^
M |= Ri (¯ x) ↔ ¬∃y, z Q(y, z) ∧
(Ri−1 (y, w) ¯ ∧ Ri−1 (z, w)) ¯
w⊂¯ ¯ x,|w|=i−2 ¯
Proof. The rightward direction follows via the fact that Age(M ) = K. To show the leftward direction, take a tuple x ¯ such that M |= ¬Ri (¯ x). By ultrahomogeneity of M, it suffices to show that x ¯ embeds into an element of K where such a y and z exist. Consider the structure A = x ¯ ∪ {a, b}, where each of a and b are Ri−1 -related to every (i − 2)-element subset of x ¯, a and b are P -related (Qrelated in the case of the second equivalence above), and no other relations hold involving a or b. It is easy to verify that this structure is in K. Definition 29. Let T = Th(M)|{P,Q,R3 } By the previous lemma, Th(M) is a definitional expansion of T . Further, from T we can in a tt way determine which definition of Ri is correct, and thus whether i − 10 ∈ S. This shows that T ≥tt S. Further, by ultra-homogeneity of T , T ≤tt K ≤tt S. It remains only to prove that the countable model of T is recursively presentable. Lemma 30. Uniformly in n, T ∩ ∃n is computable in 0n−7 . Proof. We first show that any ∃n formula in T is equivalent to a quantifier-free formula in the relations {P, Q, Ri }i≤3+n . The proof proceeds by setting up the appropriate Ehrenfeucht-Fra¨ıss´e game (see: [13], section 2.4) and seeing that ‘∃loise’ (the second player) has a winning strategy. The game is the standard EhrenfeuchtFra¨ıss´e game of length n where we start with tuples a ¯ and ¯b which have the same {P, Q, Ri }i≤3+n -quantifier-free type. Then whichever tuple c¯ ‘∀belard’ chooses, ∃loise can choose a tuple d¯ so that a ¯c¯ and ¯bd¯ satisfy the same {P, Q, Ri }i≤3+n−1 quantifier-free type. Proceeding as such, ∃loise wins the game of length n. This shows that any ∃n formula depends only on the relations {P, Q, Ri }i≤3+n ([13], Lemma 2.4.9). Thus, the ∃n formula ∃∀ . . . φ(¯ x) is equivalent to: _ ^ . . . φ(¯ x) (configurations in {P,Q,Ri }i≤3+n in K) (configurations in {P,Q,Ri }i≤3+n−1 in K)
THE DEGREES OF CATEGORICAL THEORIES WITH RECURSIVE MODELS
13
To verify whether this statement is true, we need only to be able to parse “in K” for configurations in the language {P, Q, Ri }i≤3+n . The conditions of being in K is then described recursively in 03+n−10 . We use the following case of a theorem of Knight [12] to show that the countable model of T is recursively presentable. Theorem 31. (Knight) Let T be an ℵ0 -categorical theory. If T ∩ ∃n+1 is Σ0n uniformly in n, then T has a recursive model. Since T is ℵ0 -categorical, Lemma 30 shows that T satisfies the conditions of this theorem. Thus we conclude the promised theorem. Theorem 32. Let d be a tt-degree reducible to 0(ω) . Then there exists an ℵ0 categorical theory T ∈ d in a finite language whose countable model is recursively presentable. Proof. As above, d contains a set S such that n ∈ S is uniformly recursive in 0(n) . Applying the construction to this set S yields an ℵ0 -categorical theory T ∈ d in a finite language whose countable model is recursively presentable. Corollary 33. There exists an ℵ0 -categorical theory T in a finite language whose countable model is recursively presentable and T ≡T 0(ω) . References 1. Uri Andrews, A new spectrum of recursive models using an amalgamation construction, J. Symbolic Logic 76 (2011), 883–896. 2. Uri Andrews and Alice Medvedev, Recursive spectra of strongly minimal theories satisfying the zilber trichotomy, submitted for publication. 3. C. J. Ash and J. F. Knight, Computable structures and the hyperarithmetical hierarchy, Studies in Logic and the Foundations of Mathematics, vol. 144, North-Holland Publishing Co., Amsterdam, 2000. 4. Ekaterina Fokina, Arithmetic Turing degrees and categorical theories of computable models, Mathematical logic in Asia, World Sci. Publ., Hackensack, NJ, 2006, pp. 58–69. 5. Sergei S. Goncharov and Bakhadyr Khoussainov, Complexity of theories of computable categorical models, Algebra Logika 43 (2004), no. 6, 650–665, 758–759. 6. Sergey S. Goncharov, Valentina S. Harizanov, Michael C. Laskowski, Steffen Lempp, and Charles F. D. McCoy, Trivial, strongly minimal theories are model complete after naming constants, Proc. Amer. Math. Soc. 131 (2003), no. 12, 3901–3912 (electronic). 7. Leo Harrington, Recursively presentable prime models, J. Symbolic Logic 39 (1974), 305–309. 8. Wilfrid Hodges, A shorter model theory, Cambridge University Press, Cambridge, 1997. 9. Ehud Hrushovski, A new strongly minimal set, Ann. Pure Appl. Logic 62 (1993), no. 2, 147–166, Stability in model theory, III (Trento, 1991). 10. Nazif G. Khisamiev, Strongly constructive models of a decidable theory, Izv. Akad. Nauk Kazah. SSR Ser. Fiz.-Mat. (1974), no. 1, 83–84, 94. 11. Bakhadyr Khoussainov and Antonio Montalb´ an, A computable ℵ0 -categorical structure whose theory computes true arithmetic, J. Symbolic Logic 75 (2010), no. 2, 728–740. 12. Julia F. Knight, Nonarithmetical ℵ0 -categorical theories with recursive models, J. Symbolic Logic 59 (1994), no. 1, 106–112. 13. David Marker, Model theory, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, New York, 2002, An introduction. 14. Bruno Poizat, MM. Borel, Tits, Zil0 ber et le G´ en´ eral Nonsense, J. Symbolic Logic 53 (1988), no. 1, 124–131.
